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Solve quadratic equations (ax² + bx + c = 0) instantly. Find real and complex roots, discriminant, vertex, and axis of symmetry using the quadratic formula.
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. The quadratic formula x = (−b ± √(b²−4ac)) / 2a provides its solutions.
Equation
1x² − 5x + 6 = 0
Discriminant (Δ = b² − 4ac)
1.0000000
Δ > 0 → Two real roots
Root Type
Two distinct real roots
x₁
3.0000000
x₂
2.0000000
Vertex
(2.50000, -0.250000)
Axis of Symmetry
x = 2.50000
Parabola Opens
Upward (minimum at vertex)
| Equation | Roots |
|---|---|
| x² − 1 = 0 | x = ±1 |
| x² − 4 = 0 | x = ±2 |
| x² − 5x + 6 = 0 | x = 2, 3 |
| x² + 1 = 0 | x = ±i |
| x² − 2x + 1 = 0 | x = 1 (repeated) |
| 2x² − 7x + 3 = 0 | x = 0.5, 3 |
Formula
x = (−b ± √(b² − 4ac)) / 2aa = Coefficient of x² (must be non-zero)
b = Coefficient of x
c = Constant term
b² − 4ac = Discriminant — determines the nature of the roots
Worked Example
Solve 2x² − 7x + 3 = 0
Did you know? The quadratic formula was first explicitly written by the Indian mathematician Brahmagupta around 628 AD, though Babylonian mathematicians solved quadratic equations using geometric methods as early as 2000 BC (source: MacTutor History of Mathematics).
Sources
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